 
Summary: On OneWay Functions with Optimal Locality
Benny Applebaum Yuval Ishai Eyal Kushilevitz
Computer Science Department, Technion
{abenny,yuvali,eyalk}@cs.technion.ac.il
August 18, 2005
Abstract
In [1] it was shown that, under relatively mild assumptions, there exist oneway functions (OWFs) in NC 0
4 ,
namely ones in which each bit of the output depends on just 4 input bits. This result is not far from optimal as it
is impossible to implement OWF in NC 0
2 . The gap was partially closed in [1] by showing that the existence of a
OWF in NC 0
3 is implied by the intractability of decoding a random linear code.
In this note we provide further evidence for the existence of OWF in NC 0
3 . We construct such a OWF based
on the existence of a OWF that enjoys a certain strong ``robustness'' property. Specifically, we require that the
adversary cannot invert f even if it is given, in addition to the output y = f(x), all bits of x influencing a randomly
chosen subset of the bits of y. We show that a variant of a OWF candidate suggested by Goldreich [2] satisfies
this property, assuming that it is indeed a OWF.
1 Introduction
